In these days the student of applied science finds that he can make little progress till he has acquired a knowledge of the Calculus. If he cannot acquire this knowledge sufficiently soon in the mathematical class-room, his science teacher will have to do the work which ought to fall to the duty of the mathematician, and will at the same time be thereby debarred from devoting the time to other teaching which he would like to give.

There are two reasons why the use of tangential coordinates should be introduced at an earlier stage than is customary into the teaching of analytical geometry and placed on a somewhat different footing. The educational advantages of encouraging the student to follow out a course of reasoning with the help of only a few symbols instead of grinding out hard examples by long meaningless algebra should not require much pressing, although they do not seem to be sufficiently recognised in this country; and on the other hand many theories receive their most appropriate expression in tangentials chiefly owing to the fact that the absolute has a less specialised form in tangential than in point coordinates, being (in two dimensions) the product of two distinct factors in the former and a perfect square in the latter.

The object of the following note is to derive Pascal’s theorem and the Anharmonic and Harmonic properties of Conic Sections directly from the focus and directrix definition by the early introduction of an extended form of Carnot’s theorem, viz. “the product of the ratios in which the sides of a closed polygon, taken in order, are cut by a conic = +1.” When the above have been demonstrated the simpler properties are known to be easily deducible. Subsequently the treatment of Conicoids by Pure Geometry is introduced by means of the same theorem.